If a, b, c be space described in pth, qth and rth seconds by a particle with a given velocity and moving with uniform acceleration in a straight line n what is value of a(q – r) + b(r – p) + c(p – q)?

Let, f be the uniform acceleration and u be the given initial velocity of the moving particle.
From the conditions of problem we have the following equation of motion of the particle:
u + ½(f)(2p ? 1) = a ???.(1)
u + ½(f)(2q ? 1) = b ???.(2)
u + ½(f)(2r ? 1) = c ???.(1)
Thus, a(q ? r) + b(r ? p) + c(p ? q) = [u + ½(f)(2p ? 1)](q ? r) + [u + ½(f)(2q ? 1)](r ? p) + [u + ½(f)(2r ? 1)](p ? q)
= u (q ? r + r ? p + p ? q) + f/2[(2p ? 1)(q ? r) + (2q ? 1)(r ? p) + (2r ? 1)(p ? q)] = u*0 + f/2[2(pq ? rp + qr ? pq + rp ? qr) ? q + r ? r + p ? p + q]
= f/2*0
= 0